# Properties

 Label 1440.2.bj.a Level $1440$ Weight $2$ Character orbit 1440.bj Analytic conductor $11.498$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.bj (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q+O(q^{10})$$ 48 * q $$\operatorname{Tr}(f)(q) =$$ $$48 q - 32 q^{31} - 32 q^{97}+O(q^{100})$$ 48 * q - 32 * q^31 - 32 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 0 0 −2.23604 0.0113158i 0 −0.471963 0.471963i 0 0 0
17.2 0 0 0 −2.23604 0.0113158i 0 −0.471963 0.471963i 0 0 0
17.3 0 0 0 −2.03368 + 0.929591i 0 2.49469 + 2.49469i 0 0 0
17.4 0 0 0 −2.03368 + 0.929591i 0 2.49469 + 2.49469i 0 0 0
17.5 0 0 0 −1.45381 + 1.69895i 0 −1.53029 1.53029i 0 0 0
17.6 0 0 0 −1.45381 + 1.69895i 0 −1.53029 1.53029i 0 0 0
17.7 0 0 0 −1.42597 1.72238i 0 −3.11972 3.11972i 0 0 0
17.8 0 0 0 −1.42597 1.72238i 0 −3.11972 3.11972i 0 0 0
17.9 0 0 0 −1.29263 1.82458i 0 0.306649 + 0.306649i 0 0 0
17.10 0 0 0 −1.29263 1.82458i 0 0.306649 + 0.306649i 0 0 0
17.11 0 0 0 −0.215413 + 2.22567i 0 2.32063 + 2.32063i 0 0 0
17.12 0 0 0 −0.215413 + 2.22567i 0 2.32063 + 2.32063i 0 0 0
17.13 0 0 0 0.215413 2.22567i 0 2.32063 + 2.32063i 0 0 0
17.14 0 0 0 0.215413 2.22567i 0 2.32063 + 2.32063i 0 0 0
17.15 0 0 0 1.29263 + 1.82458i 0 0.306649 + 0.306649i 0 0 0
17.16 0 0 0 1.29263 + 1.82458i 0 0.306649 + 0.306649i 0 0 0
17.17 0 0 0 1.42597 + 1.72238i 0 −3.11972 3.11972i 0 0 0
17.18 0 0 0 1.42597 + 1.72238i 0 −3.11972 3.11972i 0 0 0
17.19 0 0 0 1.45381 1.69895i 0 −1.53029 1.53029i 0 0 0
17.20 0 0 0 1.45381 1.69895i 0 −1.53029 1.53029i 0 0 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 593.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
24.h odd 2 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.bj.a 48
3.b odd 2 1 inner 1440.2.bj.a 48
4.b odd 2 1 360.2.x.a 48
5.c odd 4 1 inner 1440.2.bj.a 48
8.b even 2 1 inner 1440.2.bj.a 48
8.d odd 2 1 360.2.x.a 48
12.b even 2 1 360.2.x.a 48
15.e even 4 1 inner 1440.2.bj.a 48
20.e even 4 1 360.2.x.a 48
24.f even 2 1 360.2.x.a 48
24.h odd 2 1 inner 1440.2.bj.a 48
40.i odd 4 1 inner 1440.2.bj.a 48
40.k even 4 1 360.2.x.a 48
60.l odd 4 1 360.2.x.a 48
120.q odd 4 1 360.2.x.a 48
120.w even 4 1 inner 1440.2.bj.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.x.a 48 4.b odd 2 1
360.2.x.a 48 8.d odd 2 1
360.2.x.a 48 12.b even 2 1
360.2.x.a 48 20.e even 4 1
360.2.x.a 48 24.f even 2 1
360.2.x.a 48 40.k even 4 1
360.2.x.a 48 60.l odd 4 1
360.2.x.a 48 120.q odd 4 1
1440.2.bj.a 48 1.a even 1 1 trivial
1440.2.bj.a 48 3.b odd 2 1 inner
1440.2.bj.a 48 5.c odd 4 1 inner
1440.2.bj.a 48 8.b even 2 1 inner
1440.2.bj.a 48 15.e even 4 1 inner
1440.2.bj.a 48 24.h odd 2 1 inner
1440.2.bj.a 48 40.i odd 4 1 inner
1440.2.bj.a 48 120.w even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(1440, [\chi])$$.